I have a negation operation and two types of "logic gates" with following outputs (A, B are inputs):

"gate" 1:

A | B | OUT
0   0    1
0   1    x (no output)
1   0    x (no output)
1   1    0

"gate" 2:

A | B | OUT
0   0    x (no output)
0   1    1
1   0    0
1   1    x (no output)

I can join these two "gates" to form the third one:

A | B | OUT
0   0    1
0   1    1
1   0    0
1   1    0

Is it possible to use these (half gates or joined gates) to build regular binary logic NOR or NAND gates, given that it is possible to additionally negate any input and/or output?.

Negation with x (no output) gives no output.